Let's carefully analyze the function [tex]\( y = 3 \cdot 5^x \)[/tex] to determine its domain and range.
### Domain
The domain of a function is the set of all possible input values (x-values) that the function can accept.
For the function [tex]\( y = 3 \cdot 5^x \)[/tex]:
- Exponential functions of the form [tex]\( a \cdot b^x \)[/tex] (where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants and [tex]\( b > 0 \)[/tex]) accept any real number as the input [tex]\( x \)[/tex].
- This is because exponential functions are defined for all real values of [tex]\( x \)[/tex]; there are no restrictions on [tex]\( x \)[/tex] that would make the function undefined.
Thus, the domain of the function [tex]\( y = 3 \cdot 5^x \)[/tex] is all real numbers, which can be written as:
[tex]\[ (-\infty, \infty) \][/tex]
### Range
The range of a function is the set of all possible output values (y-values) that the function can produce.
For the function [tex]\( y = 3 \cdot 5^x \)[/tex]:
- Since [tex]\( 5^x \)[/tex] is an exponential function, [tex]\( 5^x \)[/tex] is always positive for any real number [tex]\( x \)[/tex]. Exponential functions never produce zero or negative values.
- The factor of 3 is a constant multiplier. Therefore, multiplying a positive number [tex]\( 5^x \)[/tex] by 3 still results in a positive number.
- [tex]\( 3 \cdot 5^x \)[/tex] will be positive no matter what the value of [tex]\( x \)[/tex] is.
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 5^x \)[/tex] approaches 0, but never actually reaches 0. Thus, [tex]\( 3 \cdot 5^x \)[/tex] approaches 0 but remains positive.
- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( 5^x \)[/tex] grows without bound, and so does [tex]\( 3 \cdot 5^x \)[/tex].
Therefore, the function [tex]\( y = 3 \cdot 5^x \)[/tex] produces all positive real numbers, and its range can be written as:
[tex]\[ (0, \infty) \][/tex]
In conclusion:
- The domain of [tex]\( y = 3 \cdot 5^x \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex].
- The range of [tex]\( y = 3 \cdot 5^x \)[/tex] is [tex]\( (0, \infty) \)[/tex].
